In the diagram below, there is a rectangle made of 9 squares, each of a different size. If the dimensions of the two smallest squares
in the figure are 1x1 and 4x4, can you determine the dimensions of all the other squares?
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Solution to the Problem:
The dimensions of the squares are 1x1, 4x4, 7x7, 8x8, 9x9, 10x10, 14x14, 15x15, and 18x18.
To solve, let x = the length of the segment indicated in the diagram below. Then represent the lengths of other segments in the diagram in terms of x.
Since the opposite sides of a rectangle are congruent, you can solve for x by setting the two expressions for the top and bottom sides equal to each other:
(x + 8) + (x + 12) =
(x + 3) + (x + 2) + (2x + 3)
2x + 20 = 4x + 8
2x = 12
x = 6
Now go back and substitute 6 for x in all the expressions for the sides.
Here is the complete solution.
The dimensions of the rectange are 33 x 32.
Here is Kimberly Howe's very compact solution:
Many thanks to Sharon K. Miller and Vanessa Revelli for correcting my explanation.